43 research outputs found
Covering monolithic groups with proper subgroups
Given a finite non-cyclic group , call the smallest number of
proper subgroups of needed to cover . Lucchini and Detomi conjectured
that if a nonabelian group is such that for every
non-trivial normal subgroup of then is \textit{monolithic}, meaning
that it admits a unique minimal normal subgroup. In this paper we show how this
conjecture can be attacked by the direct study of monolithic groups.Comment: I wrote this paper for the Proceedings of the conference "Ischia
Group Theory 2012" (March, 26th - 29th 2012
Covering certain monolithic groups with proper subgroups
Given a finite non-cyclic group , call the least number of
proper subgroups of needed to cover . In this paper we give lower and
upper bounds for for a group with a unique minimal normal
subgroup isomorphic to where and is cyclic. We
also show that .Comment: Communications in Algebra (2012
Covering certain Wreath Products with Proper Subgroups
For a non-cyclic finite group let be the least number of
proper subgroups of whose union is . Precise formulas or estimates are
given for for certain nonabelian finite simple groups
where is a cyclic group of order
Maximal irredundant families of minimal size in the alternating group
Let be a finite group. A family of maximal subgroups of
is called `irredundant' if its intersection is not equal to the intersection of
any proper subfamily. is called `maximal irredundant' if
is irredundant and it is not properly contained in any other
irredundant family. We denote by \mbox{Mindim}(G) the minimal size of a
maximal irredundant family of . In this paper we compute \mbox{Mindim}(G)
when is the alternating group on letters
On the number of conjugacy classes of a permutation group
We prove that any permutation group of degree has at most
conjugacy classes.Comment: 9 page
Factorizing a Finite Group into Conjugates of a Subgroup
For every non-nilpotent finite group , there exists at least one proper
subgroup such that is the setwise product of a finite number of
conjugates of . We define to be the
smallest number such that is a product, in some order, of pairwise
conjugated proper subgroups of . We prove that if is non-solvable then
while if is solvable then
can attain any integer value bigger than
, while, on the other hand, .Comment: 14 page
Covers and Normal Covers of Finite Groups
For a finite non cyclic group , let be the smallest integer
such that contains proper subgroups with the
property that every element of is contained in for some and We prove that if is a noncyclic permutation
group of degree then We then investigate the
structure of the groups with (where is
the size of a minimal cover of ) and of those with $\gamma(G)=2.